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A standard technique to lower-bound $\mathbb P(0 < X < r)$ as a function of $\mathbb E[|X|^3]$ subject to $\mathbb E[X] = 0$ and $\mathbb E[X^2] = 1$ is to consider a suitable linear combination of $| …

The obvious bound is $$\text{Pr}(X_1 + X_2 + X_3 \le x) \le \text{Pr}(X_1 \le x) \text{Pr}(X_2 \le x) \text{Pr}(X_3 \le x) \le G_1(x) G_2(x) G_3(x)$$ For something that might be slightly better, take …

If $S_i$ is the sum of the first $i$ items chosen (I assume this is with replacement), then $S_i$ has mean $\mu i$ and variance $\sigma^2 i$. Now $a(n) \le I < b(n)$ iff $S_{a(n)} \le n$ and $S_{b(n)} …

I doubt that there is a closed form in general. The characteristic function of one of your random variables is, according to Maple, ${{\rm e}^{-is}}{\rm erfc} \left( \sqrt {-is} \right)$, so the char …

I'll assume $x_0, y_0 \ge 0$. Presumably $k \ge 0$, since if $k < 0$ the only way to have $X \ge Y^k$ with $0 \le X,Y \le 1$ is $X=Y=1$. Since $X \ge Y^k$, Jensen's inequality says $E[XY] \ge E[Y^{k+1 …

Consider the following example. $W, X, Y, Z$ all have standard Cauchy distribution, but $X$ and $Y$ are independent while $Z = -W$. Then $X+Y$ also has a Cauchy distribution so its expected value doe …

Let $R(r,p,s)$, $P(r,p,s)$, $S(r,p,s)$ be the probabilities of rock, paper, scissors with initial distribution $r,p,s$, where $r + p + s$ is a power of $2$. Now for $r+p+s = 2^{n+1}$, rock will win t …

The key word is "order statistics". One thing that is known is that for any random variables $X_1,...,X_n$, if $Y_1,...,Y_n$ are the corresponding order statistics, then $\sum_i \text{Var}(Y_i) \le …

It is possible to take random variables $X_k$ and the corresponding $W_k$ so $\mathbb E[X_k] \to \infty$ while $\mathbb E[W_k]/\mathbb E[X_k] \to 0$. For example, consider $X = N$ with probability …

What might make sense is if $X$ is a function $g(U,\theta)$ of some underlying random variable $U$ (with distribution not depending on $\theta$) and $\theta$, where the differentiable function $g$ is …

Let $\mu_n$ and $\sigma_n$ be the mean and standard deviation of $X_n$. Suppose there is a sequence of positive numbers $k_n$ such that $\sum_n 1/k_n^2 < \infty$ while $\sum_n (\mu_n - k_n \sigma_n) …

Let $f(x) = \sum_{j=0}^d a_j x^j$ be any polynomial of degree $d$ such that $f(0) = 0$ and $f(x) \le 1$ for all $x \ge 0$. Then $$P(X > 0) \ge E[f(X)] = \sum_{j=0}^d a_j E[X^j]$$ Your Cauchy-Schwarz …

The CDF of $K$ is $$ P(K \le n) = P(S_n > T) = \int_{T}^\infty dt\; f_n(t)$$ The CDF of $S_K$ (for $s > T$) is $$\eqalign{P(S_K \le s) &= \sum_{n=1}^\infty P(K = n, S_n \le s) = \sum_{n=1}^\infty P(S …

As Nate Eldridge noted, the answer is no. For a positive result, you need some extra condition. Suppose e.g. the variances $\text{Var}(X_n) < c (E X_n)^2$ for $n$ sufficiently large, with some const …

Under the conditions where the Central Limit Theorem applies ($n \to \infty$ with $p$ approaching a constant), you can approximate the binomial CDF by $$ F_{n,p}(s) \approx \Phi\left(\dfrac{s-np}{\sqr …